# Boundary behaviour of $\lambda$-polyharmonic functions on regular trees

## TL;DR

This paper investigates the boundary behavior of complex $\lambda$-polyharmonic functions on regular trees, solving boundary value problems and establishing a Fatou theorem for these functions.

## Contribution

It provides new results on boundary limits and solves classical boundary value problems for $\lambda$-polyharmonic functions on regular trees.

## Key findings

- Solved Dirichlet and Riquier problems at infinity.
- Established a non-tangential Fatou theorem.
- Analyzed boundary behavior for $|\lambda|>
ho$.

## Abstract

This paper studies the boundary behaviour of $\lambda$-polyharmonic functions for the simple random walk operator on a regular tree, where $\lambda$ is complex and $|\lambda|> \rho$, the $\ell^2$-spectral radius of the random walk. In particular, subject to normalisation by spherical, resp. polyspherical functions, Dirichlet and Riquier problems at infinity are solved and a non-tangential Fatou theorem is proved.